Welcome to Issue 42 of the Secondary Magazine. It feels as if we never had a holiday doesn’t it?
The term is still quite fresh and most of the pupils still have that ’new uniform’ look about them.
What about your classroom? Have you got some new pupil work on the walls? Have you taken in
all those sets of books for the first time and made some friendly yet formative comments? We
hope this Issue will inspire you for September!
Contents
From the editor
What do you think about the new proposals for a linked pair of mathematics GCSEs? There is an
opportunity to get involved – read about it here.
Up2d8 Maths – the fourth plinth
The fortnightly Up2d8 Maths resources explore a range of mathematical themes in a topical context. This
Up2d8 resource uses the context of Anthony Gormley’s One & Other project to create ‘an astonishing
living monument’ by filling the fourth plinth in Trafalgar Square, 24 hours a day for 100 days. What would
you do?
The Interview – Jean Brennan
In this issue we hear from Jean Brennan who is director of learning in a London school – and seems to
have a soft spot for animals too.
Focus on…topology
What has the London Underground map got in common with the Seven Bridges of Königsberg? This
‘Focus on’ may provide you with the answers.
An idea for the classroom – MARS
There IS a chocolate connection with this issue’s idea but that’s only accidental! Chocolate can be a great
motivator so read on…
5 things to do
Whether you are analysing your examination results or celebrating the anniversary of the Battle of
Hastings, there is something for you here.
Diary of a subject leader – Real issues in the life of a fictional Subject Leader
Our subject leader is moving to a new job. He reflects on the changes to mathematics and the way he has
changed as a person as he prepares to make a fresh start.
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
From the editor
Had a good holiday? I was quite surprised when I got back to see a few changes in my village. There’s a
lovely thatched cottage – it’s a real chocolate box cottage, roses round the door, hollyhocks in the garden.
They’ve painted it pink! How ridiculous. A subtle cream would look nice, or an ‘off white’ perhaps, but
pink? That just spoils it – distracts your eye from the thatch. If only they’d asked me…
I braved the dreadful weather last week to go down to the beach with a friend and her children. The beach
was great – not quite blazing sun but plenty of sea, sand and ice-cream. The journey was hell. Don’t you
hate it when there is someone else in the car who’d rather be driving... mind the cyclist, watch out he’s
turning left, why don’t you park there, there’s a 30mph limit along this bit…
Are you beginning to see a connection here? I expect that there has been some of this as you have
returned to school for the new year. Have you seen what they have done to the sports hall/ the car park/
Emma’s classroom? They could have saved a lot of money by: painting it/ knocking it down/ buying a new
one. They really: are made of money/ have no common sense/ need to ask us teachers what we think.
There are plenty of occasions in everyday life when we are very quick to offer our opinions on a range of
topics but usually only to our partners or the dog. When we are asked for our professional views or our
participation in events which may influence educational policy, are we as quick to give our opinion?
It could be argued that one of the best ways to influence the future of mathematics in the UK is to become
involved in new developments and one of the newest ideas is now featured on the Qualifications and
Curriculum Development Agency (QCDA) website - the new pilot of a linked pair of mathematics GCSEs
which will be available for first teaching (in pilot centres only) from September 2010. You can read the
details on the QCDA website, but it is worth noting that:
“The philosophy behind the creation of the linked pair of GCSEs is to provide learners with a rich
experience of mathematics, enabling them to understand its importance in analysing problems related
to both the real world and to mathematics itself.”
It would be hard to argue against this philosophy, so I look forward to seeing further details of the new
qualifications. If you would like to get involved, you can:
“We will be running a number of focus groups, aimed at different stakeholder groups, over the coming
months. If you would like to attend one of these groups please let us know by emailing your full name
and contact details, as well as the particular stakeholder group you represent (eg schools, higher
education etc) to [email protected]. We will add your name to our list and invitations to the
various events will be sent out nearer the time.”
Will I see you there?
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
Up2d8 maths
The fortnightly Up2d8 Maths resources explore a range of mathematical themes in a topical context. The
resource is not intended to be a set of instructions but rather a framework which you can personalise to fit
your classroom and your learners.
This Up2d8 resource uses the context of Anthony Gormley’s One & Other project to create ‘an astonishing
living monument’ by filling the fourth plinth in Trafalgar Square, 24 hours a day for 100 days. Students are
asked what they would do if picked to spend an hour on the fourth plinth in Trafalgar Square. They are
then asked to consider two well-known songs and think about how they could be extended to occupy the
time on the plinth.
This resource is not year group specific and so will need to be read through and possibly adapted before
use. The way in which you choose to use the resource will enable your learners to access some of the Key
Processes from the Key Stage 3 Programme of Study.
Download the Up2d8 Maths resource - in PowerPoint format.
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
The Interview
Name: Jean Brennan
About you: I am the director of learning (mathematics) in an inner-city school in London.
I love sports, and animals. I have some dogs (I am not going to tell you how many) and I have two horses, one
retired and one that I ride when I get a chance. I am passionate about mathematics education and believe that
we owe it to society to make state education the best it can possibly be. I have one daughter, who is also a
teacher of mathematics. Oddly enough, I was actually a student at the school I now work in (many years ago). I
have worked in several schools, two in Dublin and four in London, I have taught girls only, boys only and both
together, all enjoyable. Teaching can be a very stressful career and the ups and downs are significant.
However, it is very rewarding work and I cannot see myself doing work outside of mathematics education.
The most recent use of mathematics in your job was... How much time have you got?! I have been working
on a new Year 7 mathematics curriculum that has rich activities and links with other subjects, culture and the
community.
Some mathematics that amazed you is... I love the connection between mathematics and art. I suppose it is
reasonable to assume that as most natural forms have a connection to the Fibonacci sequence, then this type
of ratio must be stimulating to the brain and pleasing to the eye, and I find that idea interesting – particularly
when I listen to debates about creation. However, I am fascinated by fractals and the idea of chaos theory.
Why mathematics? I did not realise this when I was younger, but it has always been mathematics! I had the
normal diet of school mathematics and the work always seemed easy but took second place to physics,
mainly due to an incredible teacher who made the concepts obvious and natural. When I started teaching, I
began to appreciate the connections between mathematics and life, and that is when my love of the subject
really started. I love a challenge – the more difficult the problem, the more determined I am to solve it, and the
day-to-day challenges of teaching mathematics provide me with plenty.
Your favourite/most significant mathematics-related anecdote is…
Things kids say at school often make me smile. It’s hard to remember one off the top of my head, but a silly
one is, when I asked a class about Pythagoras and one of them replied, “He’s a Greek guy who liked playing
with tiles.”
A maths joke that makes you laugh is… Mathematics is no joke but things I might smile at in the classroom
are: A sine curve goes off to infinity, or at least to the end of the blackboard!
Lottery: A tax on people who are bad at mathematics.
Something else that makes you laugh is…
There are holes in the sky
Where the rain gets in
But they’re ever so small
That's why the rain is thin.
- Spike Milligan.
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
Your favourite television programme is… I love Bones and all the CSI programmes, and I never miss The
Street. I also love The X Factor.
Your favourite ice-cream flavour is… I am not a lover of ice-cream but if pushed it will be vanilla.
Who inspired you? Throughout my life lots of people have inspired me and they probably do not know it.
There are many writers/mathematicians who inspire me every time I read their work. The list would probably
be too long to put here.
If you weren’t doing this job you would… I tried a few jobs before teaching and could not stand them so I
would probably be unemployed! More seriously, I have a soft spot for animals in need, so if I ever had money I
would run an animal shelter that could double up as an educational site, particularly for inner-city school kids.
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
Focus on...topology

Wolfram Alpha defines topology as the branch of pure mathematics that deals only with the
properties of a figure X that hold for every figure into which X can be transformed with a one-to-one
correspondence that is continuous in both directions. Wikipedia describes it (maybe more simply) as
a major area of mathematics concerned with spatial properties that are preserved under continuous
deformations of objects, for example deformations that involve stretching, but no tearing or gluing.

Homeomorphism could be thought of as the most basic topological equivalence in which a shape
can be continuously bent and stretched to give an apparently new shape but with the same
topological properties. Letters which are homeomorphic can be found on Wikimedia.

The term Topology was first used in print by the German mathematician Johann Benedict Listing
in 1848, though by then he had been using the word informally for up to 10 years before this.
Listing’s spelling (Topologie) was changed to the more familiar topology when the word was used
in the journal Nature in 1883.

One of the earliest and best known topological problems is the Seven Bridges of Königsberg.
Königsberg (now Kaliningrad) has seven bridges over the river Preger and the problem asks if they
can all be traversed in a single trip without doubling back, with the journey ending in the same
place it began. Euler showed that this was impossible in 1736 Euler realised that, in order to make
the journey without doubling back, a traveller in the middle of the journey must enter a land mass
via one bridge and leave by another, meaning that land mass must have an even number of
connecting bridges. If the traveller, at the start of the journey, leaves one land mass, then a single
bridge will suffice and upon completing the journey the traveller may again only require a single
bridge to reach the ending point of the journey. The starting and ending points then, are allowed
to have an odd number of bridges. But if the starting and ending point are to be the same land
mass, then it and all other land masses must have an even number of connecting bridges. Since all
of the land masses of Königsberg have an odd number of connecting bridges and the journey that
would take a traveller across all the bridges, one and only one time during the journey, proves to
be impossible.

The Alvin Ailey Dance Theatre in New York have created a piece, Memoria, which explores
topology. One reviewer said, on exploredance.com, “The structure I saw in Memoria was a
mathematical function known as a topology. A topology is a geometrical set that can be stretched in
space without being torn apart. For instance, if one started with a donut with one hole, the donut could
be made larger or smaller, twisted or reshaped from circular to elliptical and it would still be the same
topology even though at first glance each variant might look very different. On the other hand, if one
punched a second hole in the donut, it would no longer be the same topology.
“Memoria uses three dancers as points of reference. The triangle formed by the dancers reverses
orientation, compacts and shifts, but is never torn apart. There is a section with a solo and a duet, but
then the triangle reasserts itself. The other dancers form a second set that moves around and between
the first set. Sometimes the sets use different movements and sometimes similar movements. The sets
are also delineated by different costumes.”
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
An idea for the classroom – MARS
Having been inspired by Malcolm Swan’s work in the resource Improving learning in mathematics:
challenges and strategies, I was interested to find details of another project which involves his work. MARS
(Mathematics Assessment Resource Service) is a joint project between Michigan State University,
University of California at Berkeley and the Shell Centre at Nottingham University. On the website, there
are some sample assessment tasks which interested me. I have described two tasks here but do go and
look at some of the others on the site.
Here is an idea for a task using supermarket trolleys
Pupils are familiar with the trolley and will have had experience getting a trolley, pushing it round the
supermarket and stacking it in the car park. Using this familiarity, pupils can then try to solve problems
like:
‘Create a rule that will tell you the length of storage space (S) needed when all you know is the number of
supermarket carts to be stored.’
Another task is about chocolate polyhedra.
You have a mould shaped like a cube which you half fill with chocolate, but you then leave the chocolate
to solidify in different positions, for example suspended by one of the vertices.
What are the shapes of the chocolates that are produced?
There is a short version and a long version of each task on the MARS website, which includes some pupil
solutions and some answers – for this task the pictures would seem to be useful, unless you are willing to
try some practical work…
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
5 things to do this fortnight

On 2 November, Gresham College is hosting Mathematics, motion, and truth: the Earth goes
round the Sun at Barnard’s Inn Hall. The reality of the Earth's motion, as proclaimed by Copernicus,
quickly proved contentious. Accepted by Kepler, disputed by theologians (Lutheran and Catholic
alike), veiled in suggestions of mere convenience, adopted and explained by Newton as a
consequence of universal gravitation, parent of the notion of force – What is involved in accepting
as true that the Earth goes round the Sun? This lecture traces these debates from the early 1600s
to the time of Poincaré. More details are available on the Gresham College website.

Lesson Study means many things to many people, but generally it describes a process where
teachers plan, teach and develop lessons collaboratively. Lesson Study is a key feature of the way
many teachers in Japan work, and has now been adopted in American and, more recently, in some
English schools.
The NCETM is planning a national NCETM Lesson Study Project which will begin with a launch
seminar for 15 schools or colleges on Wednesday 14 October 2009. Applications to join the project
must be in by 21st September.

Have you analysed your exam results yet? A number of the exam boards have a free tool which
gives a question level breakdown, such as Edexcel’s ResultsPlus and AQA’s Enhanced Results
Analysis. These tools not only allow easy identification of students who might benefit from a
retake, but also allow you to see which areas of the curriculum your Year 11s found most
challenging which might allow you to review the teaching and learning of these topics with your
new Year 11.

How do you feel about Functional Skills? Have a look at this video and booklet from the QCDA
[formerly QCA] which gives case studies and details lessons learned from the pilot centres.

How will you be celebrating the 943rd anniversary of the Battle of Hastings? Hastings Week takes
place from 10 to 18 October 2009.
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching
Diary of a subject leader
Real issues in the life of a fictional Subject Leader
With a combination of joy and sadness, I am moving on and leaving my present post as subject leader. I
have secured a job in a larger school, providing me with an opportunity to lead a bigger department,
teach A Level and further my experience.
Inevitably, such a move prompts me to reflect on the successes and frustrations over the past years. Much
has changed. The most notable is the shift towards a process-driven curriculum, forcing many of my staff
to reflect upon their own practice. In addition, the loss of SATs, coursework and the intermediate tier has
made us ask further questions and look for alternative approaches.
On a personal level, I am no longer the naïve, inexperienced head of department who arrived at the school
some years back. Careful management of people and resources was required in order for the department
and its staff to develop successfully. I think back to crass decisions that I made and failed initiatives that I
implemented with some embarrassment but little regret. I am a great believer in the cliché that mistakes
need to be made in order to make progress. I just hope I did not make too many at the expense of others.
So what am I taking to my new post? One thing that has become increasingly apparent to me in recent is
years is the power and influence a head of mathematics can have within a school. This does however
come with increased responsibility. I now understand that the department must function as a team and
that it is the role of the subject leader to encourage contributions from all towards a shared vision. I now
understand and have experienced how strong and effective leadership can raise standards.
I will inevitably make mistakes in my new post, especially at the start. This time however, I am approaching
the job with a different outlook. I feel that I now understand what the post is and what it entails. No longer
am I one of the young teachers laughing and joking in the staffroom. I have become one of the middleaged, ‘seen it – done it’ mob which once seemed a long way into the future. I am still not entirely sure as
to when that transition happened, but I know it did.
.www.ncetm.org.uk
A Department for Children, Schools and Families initiative to
enhance professional development across mathematics teaching